“THOSE OLD-FASHIONED SYSTEMS really get me,” said Simon Simpson, shaking his head. “Do you know, for example, the weird system of Cudworth?”
“Yes,” said the Sorcerer, “Professor Quincy showed it to us yesterday.”
“Now, if that isn’t the craziest system I’ve ever seen in my life! To go to all that trouble just to get a normalizer! Sometimes I wonder about the sanity of some of my colleagues!”
“Oh, I don’t know,” said the Sorcerer. “I found it quite an interesting challenge to find the key to the system.”
“But the system is so unnecessarily complicated!” said Simpson. “Cudworth seems to delight in making things as difficult as possible. My philosophy is the very opposite—I like to make things as simple as possible.
“Actually, Quincy’s system is not all that bad,” Simpson went on, “and considering that it was devised before one-sided quotation was discovered, it is understandable. Still, the last rule—the erasure rule—seems artificial and nothing more than an ad hoc device to remedy a none-too-good situation. But even here, Quincy could have done better: he could have replaced the two rules—Rule M and Rule K—by a single rule: if x names y, then Lx names yQ2. That one rule would have yielded the double and triple fixed-point principles, and all the problems that he probably gave you would be solvable.
“Of course, the Roberts system is much cleaner and more natural. But even this system is more complex than it need be for the problems that really matter from a practical point of view. I am not interested in such academic questions as whether an x can be found that creates its own repeat; of what significance is that to robotry? My approach is purely pragmatic. I’m interested only in questions of sociological importance: Which robots create which? Which destroy which? Which are friends, best friends, enemies, worst enemies, of which? And for matters like these, my program system is the most efficient of all.”
“Do you use one-sided or two-sided quotation?” asked the Sorcerer.
“Neither; my system is quotationless. I don’t bother with certain expressions naming others.”
“That’s interesting,” said the Sorcerer. “I have also been experimenting of late with quotationless systems—not for robotry, but for certain general problems that arise with self-reference. I am really curious to hear about your system.”
“My rules are direct, short, sweet, and to the point. I use the symbols C, D, F, E, Ċ, Ḋ, Ḟ, Ė and my rules are these:
Rule C. Cx creates x.
Rule Ċ. Ċx creates xx.
Rule D. Dx destroys x.
Rule Ḋ. Ḋx destroys xx.
Rule F. Fx is the best friend of x.
Rule Ḟ. Ḟx is a friend of xx.
Rule E. Ex is the worst enemy of x.
Rule Ė. Ėx is an enemy of xx.
“What could be more direct than these? The solutions to all the problems I will now give you are shorter than any you have yet seen. For example, it is obvious that a self-reproducing robot is ĊĊ and a self-destroying one is ḊḊ. So here are some problems, and I am sure you will solve them quite easily. And incidentally, the solutions place the solutions in Roberts’s system in a clearer light, in a manner I will later explain. But for now, let’s concentrate on just the problems.” Professor Simpson’s problems follow.
• 1 •
Find a distinct x and y such that each creates the other.
• 2 •
Find an x and y such that x creates y and y destroys x. There are two solutions.
• 3 •
Show that for any expression a there is some x that creates ax, and some x that destroys ax.
• 4 •
Given any expressions a and b, show:
(a) There are an x and y such that x creates ay and y creates bx. (There are two solutions.)
(b) There are an x and y such that x destroys ay and y destroys bx. (Two solutions.)
(c) There are an x and y such that x creates ay and y destroys bx.
• 5 •
Find an x that is a friend of itself.
• 6 •
Find an x that creates its best friend.
• 7 •
Find an x that creates a friend that is not its best friend.
• 8 •
Find an x that is a friend of its worst enemy.
• 9 •
Find an x that is the best friend of one of its enemies.
• 10 •
Find an x and y such that x creates the best friend of y and y destroys the worst enemy of x.
• 11 •
Find an x that is the best friend of one that destroys its worst enemy.
Find an x that creates some y that is a friend of some z that is the worst enemy of some w that destroys the best friend of the worst enemy of x.
“And so you see,” said Simpson proudly, “that all sorts of complicated sociological situations can be programmed very easily in my system.”
“Your system is indeed neat and economical,” said the Sorcerer, “and I like it very much. It has many similarities to a system of mine, and by a strange coincidence you use the dot over a letter in much the way I do.
“One thing you may not realize is that all your solutions can be easily transformed into solutions in the Roberts system simply by replacing C by CQ, Ċ by CRQ, D by DCQ, Ḋ by DCRQ, F by FCQ, Ḟ by FCRQ, E by ECQ, and Ė by ERCQ. For example, your x that creates itself is ĊĊ. If we replace Ċ by CRQ, we get CRQCRQ, which is the x of the Roberts system that creates itself. The same holds for all your solutions.”
“I certainly do realize that,” said Simpson, “and that is what I meant before when I said that my solutions put the Roberts solutions in a clearer light.”
The reader can easily check that in each of the twelve problems above, the solutions in the Simpson system can be transformed into solutions in the Roberts system by just the method stated by the Sorcerer.
1. ĊCĊ and CĊCĊ
2. Solution 1: x = ĊDĊ, y = DĊDĊ
Solution 2: x = CḊCḊ, y = ḊCḊ
3. An x that creates ax is ĊaĊ. An x that destroys ax is ḊaḊ.
4. (a) Solution 1: x = ĊaCbĊ, y = CbĊaCbĊ
Solution 2: x = CaĊbCaĊ, y = ĊbCaĊ
(b) Same, using Ḋ in place of Ċ and D in place of C
(c) Solution 1: x = ĊaDbĊ, y = DbĊaDbĊ
Solution 2: x = CaḊbCaḊ, y = ḊbCaḊ
5. ḞḞ
6. ĊFĊ
7. CḞCḞ
8. ḞEḞ
9. FĖFĖ (Robot ĖFĖ is satanic)
10. One Solution: x = ĊFDEĊ, y = DEĊFDEĊ
Another Solution: x = CFḊECFḊ, y = ḊECFḊ
11. x = FḊEFḊ
12. There are several solutions. One of them is x = ĊFEDFEĊ (which creates FEDFEx). Another is x = CḞEDFECḞ. Another is CFEḊFECFEḊ.
In each solution, the y, z, and w can easily be found.